#497795

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
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	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404991);