#765948

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

3012 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)(11,15)(14,16)(17,18,22,20)(19,23,25,26)(21,28,30,24)(27,32,29,31);(5,6,7,8)(10,13)(14,16)(18,20)(19,25)(24,28)(27,29);(1,2,3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31);(9,11)(10,14)(12,15)(13,16)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31);(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

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, 673144);