#861532

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1981 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=gi,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 768728);