#360688

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,14,18,15,19,13,28)(11,16,21,33,22,32,29,31)(12,23,27,34,26,38,25,17)(20,35,36,30,24,37,39,40);(2,5)(6,7)(9,11)(10,16)(12,24)(13,29)(14,21)(15,22)(17,30)(18,33)(19,32)(20,26)(23,37)(25,36)(27,39)(28,31)(34,40)(35,38);(9,12,13,25,15,26,14,27)(10,17,18,23,19,34,28,38)(11,20,21,36,22,24,29,39)(16,30,31,35,32,40,33,37);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267884);