#498543

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
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

756 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)(9,10,13,18,14,16,21,23,12,17,15,22,11,19,20,24);(6,7)(9,10,15,22,14,16,20,24,12,17,13,18,11,19,21,23);(1,2,3,4)(9,11,12,14)(10,16,17,19)(13,20,15,21)(18,23,22,24);(5,7,8,6)(9,11,12,14)(10,16,17,19)(13,20,15,21)(18,23,22,24)

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,d^a=di,c^b=ci,d^b=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405739);