#375572

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=gh,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282768);