#387196

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

906 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

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, 294392);