#323254

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

791 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=i,d^2=eh,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, 230450);