#207751

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

886 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)(3,6)(4,5)(7,8)(9,10)(11,14,18,22)(12,19)(13,16)(15,20)(17,23,24,21)(25,26,29,32,28,31,35,38)(27,30,34,37,33,36,39,40);(9,11,13,18)(10,14)(12,17,20,24)(15,21)(16,22)(19,23)(25,27,29,34,28,33,35,39)(26,30,32,37,31,36,38,40);(2,5)(6,8)(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,21,22,23)(26,31)(30,36)(32,38)(37,40);(1,3,4,7)(2,6,5,8)(27,33)(30,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=fhi,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, 114947);