#323203

order	512
nilpotency class	3
isoclinism	isoclinic class of #323197
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

779 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=h,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230399);