#495651

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×Q₈	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

936 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)(9,10,14,18,15,19,30,28)(11,20,23,21,24,13,17,29)(12,16,27,33,25,31,38,39)(22,34,37,35,36,26,32,40);(9,11,21,10,15,24,29,19)(12,22,35,16,25,36,40,31)(13,28,14,17,20,18,30,23)(26,39,27,32,34,33,38,37);(2,5)(6,8)(9,12)(10,16)(11,22)(13,26)(14,27)(15,25)(17,32)(18,33)(19,31)(20,34)(21,35)(23,37)(24,36)(28,39)(29,40)(30,38);(1,3,4,7)(2,6,5,8)(12,25)(16,31)(22,36)(26,34)(27,38)(32,37)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=f,e^2=i,g^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, 402847);