#777335

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

4608 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)(5,6,7,8)(9,10,12,15)(11,14,16,13)(17,18)(19,23)(20,21)(22,24);(9,11,12,16)(10,13,15,14)(17,19,21,24)(18,22,20,23);(5,6)(7,8)(17,19)(18,22)(20,23)(21,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,20,21,18)(19,23,24,22);(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 684531);