#758214

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

1874 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)(10,13)(14,15)(17,18,22,21)(19,24,20,23)(25,26,29,27)(28,30,32,31);(9,10)(11,15)(12,13)(14,16)(17,18,22,21)(19,23,20,24)(25,27,29,26)(28,30,32,31);(10,13)(14,15)(17,19,22,20)(18,23,21,24)(25,28)(26,30)(27,31)(29,32);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,24,21,23);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,21,22,18)(19,24,20,23)

Polycyclic presentation:

2^9|a^2=hi,b^2=fgh,c^2=g,d^2=fi,e^2=fi,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, 665410);