#6553251

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,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, 6460447);