#877238

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=i,e^2=g,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, 784434);