#6368917

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

1545 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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	#72306705, #72306708, #72306716
maximal normal subgroup of	#72306705, #72306708, #72306716

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276113);