#206915

order	512
nilpotency class	3
isoclinism	isoclinic class of #195990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

906 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=eh,g^b=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114111);