#312187

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
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

588 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)(17,18,21,25,22,26,31,32)(19,27,28,24,20,29,30,23);(2,5)(6,7)(10,14)(11,12)(15,16)(17,19,22,20)(18,23,26,24)(21,30,31,28)(25,29,32,27);(9,10)(11,14)(12,16)(13,15)(17,20,22,19)(18,24,26,23)(21,28,31,30)(25,27,32,29);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,14,15,16)(18,25)(21,31)(23,29)(24,27)(26,32)(28,30)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219383);