#393202

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 300398);