#299668

order	512
nilpotency class	3
isoclinism	isoclinic class of #299653
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1063 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206864);