#494994

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 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)(9,10,13,18,15,21,29,19)(11,22,24,16,12,26,23,34)(14,30,28,20,32,37,40,38)(17,31,36,27,33,39,35,25);(2,5)(6,8)(9,11,15,12)(10,16)(13,24,29,23)(14,27,32,25)(17,20)(18,26)(19,22)(21,34)(28,39,40,31)(30,35)(33,38)(36,37);(9,12,15,11)(10,17,21,33)(13,24,29,23)(14,31,32,39)(16,20,34,38)(18,35,19,36)(22,37,26,30)(25,40,27,28);(1,3,4,7)(2,6,5,8)(9,13,15,29)(10,18,21,19)(11,23,12,24)(14,28,32,40)(16,22,34,26)(17,35,33,36)(20,37,38,30)(25,39,27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=eh,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi

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, 402190);