#265354

order	512
nilpotency class	3
isoclinism	isoclinic class of #265339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ehi,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=gh

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, 172550);