#493654

order	512
nilpotency class	4
isoclinism	isoclinic class of #493653
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20609
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#63048	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2169 conjugacy classes of subgroups; only the 304 normal subgroups are shown.

normal subgroups only

This group has 2169 conjugacy classes of subgroups — too many to draw. Only its 304 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)(3,15,19,37)(4,20,24,40)(5,12,6,11)(7,13)(8,14)(9,17)(10,23,35,29)(16,44)(18,43,28,57)(21,53)(22,52)(25,39,27,36)(26,33)(30,41)(31,58,60,64)(32,48)(34,54)(38,56)(42,49)(45,51,55,61)(46,59)(47,63)(50,62);(1,3)(2,9)(4,16)(5,17)(6,26)(7,28)(8,19)(10,31)(11,15)(12,37)(13,32)(14,33)(18,30)(20,51)(21,45)(22,55)(23,49)(24,47)(25,50)(27,48)(29,46)(34,42)(35,60)(36,43)(38,59)(39,57)(40,61)(41,62)(44,52)(53,63)(54,64)(56,58);(1,4)(2,10)(3,16)(5,21)(6,22)(7,29)(8,24)(9,31)(11,34)(12,38)(13,40)(14,35)(15,42)(17,45)(18,49)(19,47)(20,41)(23,30)(25,56)(26,55)(27,54)(28,46)(32,61)(33,60)(36,53)(37,59)(39,52)(43,63)(44,57)(48,64)(50,58)(51,62);(1,5,8,6)(2,11,14,12)(3,17,19,26)(4,21,24,22)(7,25,30,27)(9,15,33,37)(10,34,35,38)(13,36,41,39)(16,45,47,55)(18,48,28,50)(20,52,40,53)(23,54,29,56)(31,42,60,59)(32,43,62,57)(44,61,63,51)(46,58,49,64)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303641, #72304193
maximal normal subgroup of	#72303641, #72304193
maximal quotient of	#72302892, #72324280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400850);