#758746

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

1852 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=i,e^2=fhi,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, 665942);