#411034

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

948 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)(19,21)(20,22)(23,24)(25,26,29,33)(27,34,35,31)(28,30,37,32)(36,39,40,38);(9,10,12,14)(11,15,16,13)(17,19,20,23)(18,21,22,24)(26,30)(28,37)(31,38)(32,33)(34,39)(36,40);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(25,27)(26,31)(28,36)(29,35)(30,38)(32,39)(33,34)(37,40);(1,3,4,7)(2,6,5,8)(18,22)(21,24)(27,35)(31,34)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,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, 318230);