#763508

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2476 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=g,e^2=fg,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, 670704);