#726186

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1816 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,16,15)(19,23)(22,24)(26,30)(31,32);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,20)(19,22,23,24);(1,3,5,8)(2,4,6,7)(9,10,12,14)(11,13,16,15)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27,29,28)(26,31,30,32);(1,4,5,7)(2,3,6,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,24,23,22)(25,28,29,27)(26,32,30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 633382);