#6672947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3488	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580143);