#6368923

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

1618 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)(17,18,20,25)(19,24,27,22)(21,23,29,26)(28,31,32,30);(1,2)(3,9)(4,14)(5,8)(6,11)(7,12)(10,15)(13,16)(17,19,20,27)(18,22,25,24)(21,28,29,32)(23,30,26,31);(1,3,7,4)(2,9,12,14)(5,15,6,13)(8,10,11,16)(18,23)(21,29)(22,30)(24,31)(25,26)(28,32);(1,4,7,3)(2,10,12,16)(5,15,6,13)(8,9,11,14)(17,19)(18,24)(20,27)(21,28)(22,25)(23,31)(26,30)(29,32);(1,5,7,6)(2,11,12,8)(3,13,4,15)(9,10,14,16)(19,27)(22,24)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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	#72306700, #72306708, #72306724
maximal normal subgroup of	#72306700, #72306708, #72306724

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276119);