#412538

order	512
nilpotency class	3
isoclinism	isoclinic class of #393904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319734);