#309267

order	512
nilpotency class	3
isoclinism	isoclinic class of #264427
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

556 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=f,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=dh,g^c=gi

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, 216463);