#496098

order	512
nilpotency class	3
isoclinism	isoclinic class of #23414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#127
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21683
prime radical	#23794	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 204 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ch,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224962
maximal normal subgroup of	#72224962

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403294);