#278348

order	512
nilpotency class	3
isoclinism	isoclinic class of #278339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,g^b=gh,f^c=fi,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, 185544);