#7043370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6950566);