#7043495

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

1275 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)(19,26)(20,29)(21,24)(22,25)(23,31)(27,32)(28,30);(1,3,5,8)(2,6,4,7)(9,11,14,10)(12,15,13,16)(17,19)(18,24)(20,30)(21,22)(23,28)(25,26)(27,31)(29,32);(9,12)(10,15)(11,16)(13,14)(17,20,29,18,22,31,23,25)(19,27,32,26,21,30,28,24);(1,4,5,2)(3,6,8,7)(9,13,14,12)(10,16,11,15)(17,21,22,19)(18,24,25,26)(20,30,31,27)(23,32,29,28);(10,11)(15,16)(18,25)(19,21)(20,31)(28,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6950691);