#6547354

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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)(10,13)(14,16)(17,18,23,25,22,26,21,24)(19,27,28,32,30,20,29,31);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,29)(22,30)(23,28)(24,32)(25,31)(26,27);(10,13)(14,16)(17,20)(18,19)(21,31)(22,27)(23,32)(24,28)(25,29)(26,30);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(17,21,22,23)(18,24,26,25)(19,28,30,29)(20,31,27,32);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,26,24)(19,29,30,28)(20,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6454550);