#6369784

order	512
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×C₄²
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#376
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2673 conjugacy classes of subgroups; only the 1808 normal subgroups are shown.

normal subgroups only

This group has 2673 conjugacy classes of subgroups — too many to draw. Only its 1808 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,13,17,35)(4,9)(5,10)(6,11)(7,12)(8,16,26,41)(14,33,38,54)(15,34,39,55)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,37,47,58)(25,40,48,59)(36,53,56,63)(42,49)(43,50)(44,51)(45,52)(46,57,61,64)(60,62);(1,3)(2,8,12,26)(4,14)(5,15)(6,16)(7,17)(9,24,29,47)(10,25,30,48)(11,13,32,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(27,46,49,61)(28,33,51,54)(31,34,52,55)(42,56)(43,57)(44,58)(45,59)(50,53,62,63)(60,64);(1,4,6,19,7,20,23,44)(2,9,11,28,12,29,32,51)(3,14,16,37,17,38,41,58)(5,18,22,43,21,42,45,60)(8,24,13,33,26,47,35,54)(10,27,31,50,30,49,52,62)(15,36,40,57,39,56,59,64)(25,46,34,53,48,61,55,63);(1,5)(2,10)(3,15)(4,18)(6,22)(7,21)(8,25)(9,27)(11,31)(12,30)(13,34)(14,36)(16,40)(17,39)(19,43)(20,42)(23,45)(24,46)(26,48)(28,50)(29,49)(32,52)(33,53)(35,55)(37,57)(38,56)(41,59)(44,60)(47,61)(51,62)(54,63)(58,64);(5,21)(10,30)(15,39)(18,42)(22,45)(25,48)(27,49)(31,52)(34,55)(36,56)(40,59)(43,60)(46,61)(50,62)(53,63)(57,64)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,f^2=i,b^a=bf,h^a=hi,g^b=gi,e^d=ei

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285922, #72286055, C₂×#6369784
maximal normal subgroup of	#72285922, #72286055, C₂×#6369784
maximal quotient of	C₂×#6369784

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276980);