#505185

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33524	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412381);