#498542

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×#127	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de

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, 405738);