#393812

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1838 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,h^b=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301008);