#322443

order	512
nilpotency class	3
isoclinism	isoclinic class of #322409
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229639);