#393171

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1206 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)(5,6,8,7)(9,10)(11,19)(12,23)(13,26)(14,16)(15,21)(17,25)(18,27)(20,33)(22,31)(24,30)(28,36)(29,40)(32,35)(34,37)(38,39);(5,7,8,6)(9,11,13,21,14,22,27,36)(10,15,18,19,16,28,26,31)(12,20,24,34,25,35,39,40)(17,29,30,32,23,37,38,33);(1,2,3,4)(10,16)(15,28)(17,23)(18,26)(19,31)(29,37)(30,38)(32,33);(9,12,13,24,14,25,27,39)(10,17,18,30,16,23,26,38)(11,20,21,34,22,35,36,40)(15,29,19,32,28,37,31,33)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,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, 300367);