#8702673

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8609869);