#498110

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂×#120	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=dei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405306);