#397177

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1480 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304373);