#6501013

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1917 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)(3,6)(4,5)(7,8)(18,22)(19,23)(24,28)(25,30)(26,29)(31,32);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,26,29)(20,24,27,28)(25,30,31,32);(9,11)(10,13)(12,16)(14,15)(17,19,21,26)(18,23,22,29)(20,25,27,31)(24,30,28,32);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,24)(19,25)(21,27)(22,28)(23,30)(26,31)(29,32);(10,14)(13,15)(19,26)(20,27)(23,29)(24,28)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282026, #72282545, #72282699
maximal normal subgroup of	#72282026, #72282545, #72282699

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6408209);