#415724

order	512
nilpotency class	3
isoclinism	isoclinic class of #394183
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 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)(3,6)(4,5)(7,8)(17,18,20,22)(19,21,23,24)(25,26)(27,35)(28,31)(29,39)(30,34)(32,37)(33,36)(38,40);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24)(25,27,36,38,30,37,29,28)(26,31,39,32,34,40,33,35);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,23)(21,24)(25,28,29,37,30,38,36,27)(26,32,33,31,34,35,39,40);(9,11,12,15)(10,14,13,16)(18,22)(21,24)(27,37)(28,38)(31,40)(32,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=e,d^2=gh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,d^c=di

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, 322920);