#1428645

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

3320 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=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, 1335841);