#6469078

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#34963
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,16)(4,19)(5,10)(6,12)(7,14)(8,13)(9,15)(11,22)(17,32)(18,29)(20,26)(21,30)(23,27)(24,28)(25,31);(1,3)(2,10)(4,20)(5,8)(6,17)(7,24)(9,18)(11,26)(12,27)(13,29)(14,16)(15,28)(19,30)(21,23)(22,25)(31,32);(1,4,8,22,9,17,7,23)(2,11,14,19,15,27,13,32)(3,6,5,21,18,20,24,25)(10,26,16,30,28,12,29,31);(1,5)(2,10)(3,7)(4,21)(6,23)(8,18)(9,24)(11,26)(12,27)(13,29)(14,16)(15,28)(17,25)(19,30)(20,22)(31,32);(1,6,7,25,9,20,8,21)(2,12,14,31,15,26,13,30)(3,17,24,22,18,4,5,23)(10,27,16,32,28,11,29,19)

Polycyclic presentation:

2^9|c^2=g,e^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,f^c=fi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72217290, #72224957, …
maximal normal subgroup of	#72217287, #72217290, #72224957, …
maximal quotient of	C₂×#6469078

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376274);