#758020

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3672 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,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, 665216);