#299729

order	512
nilpotency class	3
isoclinism	isoclinic class of #299717
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

1013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=eh,e^2=h,f^2=i,g^2=h,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^b=gh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206925);