#362339

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 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)(3,6)(4,5)(7,8)(9,10,16,22,14,20,15,21)(11,23,27,38,26,17,31,34)(12,19,29,37,13,18,30,36)(24,32,25,33,40,39,28,35);(9,11,15,27,14,26,16,31)(10,17,21,34,20,23,22,38)(12,28,29,40,13,25,30,24)(18,35,36,39,19,33,37,32);(2,5)(6,8)(9,12,14,13)(10,18,20,19)(11,24,26,40)(15,30,16,29)(17,32,23,39)(21,37,22,36)(25,31,28,27)(33,38,35,34);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,19,20,18)(11,25,26,28)(15,30,16,29)(17,33,23,35)(21,37,22,36)(24,31,40,27)(32,38,39,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269535);