#6368876

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

1551 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^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	#72306700, #72306707, #72306722
maximal normal subgroup of	#72306700, #72306707, #72306722

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276072);