#6357457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357427
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fh,c^2=g,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306799, #72306808, #72306812, …
maximal normal subgroup of	#72306799, #72306808, #72306812, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264653);