#7622628

order	512
nilpotency class	4
isoclinism	isoclinic class of #62636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂³×D₈
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂²×C₄×D₈
prime radical	#62635	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#131 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1885 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72217290, #72224948, …
maximal normal subgroup of	#72217287, #72217290, #72224948, …
maximal quotient of	#72232006, #72232015, #72232027, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529824);