#1258756

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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,e^2=g,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, 1165952);