#496329

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×Q₈	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×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1060 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)(9,10,15,18,13,19,12,25)(11,21,23,31,22,16,26,32)(14,20,27,34,28,17,29,35)(24,33,38,40,36,30,37,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,26)(13,22)(14,24)(15,23)(17,33)(18,32)(19,21)(20,30)(25,31)(27,38)(28,36)(29,37)(34,39)(35,40);(3,8)(5,7)(10,17)(11,22)(14,28)(16,30)(18,34)(19,20)(21,33)(23,26)(25,35)(27,29)(31,39)(32,40);(2,6)(5,7)(9,12,13,15)(10,18,19,25)(11,23,22,26)(14,27,28,29)(16,31,21,32)(17,34,20,35)(24,37,36,38)(30,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,f^2=i,g^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, 403525);