#498029

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=deh,g^b=gi,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, 405225);