#325955

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=fh,d^2=ehi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233151);