#619940

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1920 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

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, 527136);