#311353

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=e,d^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 218549);