#1259543

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3353 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1166739);