#322378

order	512
nilpotency class	3
isoclinism	isoclinic class of #322377
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

866 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eg,d^2=e,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=dh,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, 229574);