#759858

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
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

2558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 24 points:

(2,4)(9,10,13,15)(11,16,12,14)(19,23)(20,24);(5,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,21)(19,20,23,24);(5,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(9,12,13,11)(10,14,15,16)(17,20,22,24)(18,23,21,19);(9,10)(11,16)(12,14)(13,15)(17,21,22,18)(19,24,23,20)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 667054);