#496317

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1098 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,24)(13,18)(14,19,28,16)(15,21)(17,26)(22,30)(23,31,35,29)(25,33)(27,32,38,34)(36,39,37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,25)(13,21)(14,23)(16,31)(17,33)(18,20)(19,29)(22,26)(24,30)(27,37)(28,35)(32,39)(34,40)(36,38);(3,8)(5,7)(10,16)(11,21)(14,28)(15,29)(17,32)(18,19)(20,31)(22,25)(24,34)(27,38)(30,39)(33,40);(2,6)(5,7)(9,12,13,26)(10,17,18,24)(11,22,21,25)(14,27,28,38)(15,30,20,33)(16,32,19,34)(23,36,35,37)(29,39,31,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403513);