#206902

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
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

890 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)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22)(25,26,28,32,29,31,36,37)(27,30,34,39,33,38,35,40);(2,5)(6,7)(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23)(25,27,28,35,29,33,36,34)(26,30,37,39,31,38,32,40);(1,3,4,8)(2,6,5,7)(27,33)(30,38)(34,35)(39,40);(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)(26,31)(30,38)(32,37)(39,40)

Polycyclic presentation:

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

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, 114098);