#6545498

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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
prime radical	#23850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1680 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)(3,8)(4,6)(5,7)(17,18,23,31,22,24,21,27)(19,28,29,25,20,26,30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,20)(27,31)(29,30);(10,13)(14,16)(17,19,22,20)(18,25,24,32)(21,29,23,30)(26,31,28,27);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,26,24,28)(21,29,23,30)(25,31,32,27);(11,15)(14,16)(17,21,22,23)(18,27,24,31)(19,29,20,30)(25,26,32,28)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452694);