#8226770

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#59090	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8133966);