#6548300

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	#24441	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1581 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=ei,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, 6455496);