#758208

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

1860 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)(9,10,13,12)(11,15,16,14)(18,21)(22,23)(25,26,30,29)(27,32,28,31);(10,12)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26,30,29)(27,31,28,32);(11,16)(14,15)(17,18,20,21)(19,22,24,23)(25,27,30,28)(26,31,29,32);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,27)(26,32,29,31);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,19,20,24)(18,22,21,23)(25,29,30,26)(27,32,28,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=fi,e^2=i,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, 665404);