#495646

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=fi,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402842);