#6471365

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308886, #72308919, #72308967
maximal normal subgroup of	#72308886, #72308919, #72308967

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6378561);