#8701683

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608879);