#758148

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2471 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,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, 665344);