#392019

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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

802 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)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,19)(18,21)(20,23)(22,24)(25,27,30,38,29,37,28,36)(26,31,34,40,33,35,32,39);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,23)(21,24)(25,28,29,30)(26,32,33,34)(27,36,37,38)(31,39,35,40);(2,5)(3,7)(9,11,12,15)(10,14,13,16)(26,33)(31,35)(32,34)(39,40)

Polycyclic presentation:

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

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, 299215);