#6368959

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1556 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306601, #72306703, #72306710
maximal normal subgroup of	#72306601, #72306703, #72306710

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276155);