#493886

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1806 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di,f^c=fi,g^c=ghi,g^f=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, 401082);