#299553

order	512
nilpotency class	3
isoclinism	isoclinic class of #299541
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

1100 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)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24);(3,6)(4,8)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24);(1,3,4,2,5,6,8,7);(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,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,f^b=fi,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206749);