#8970107

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

4470 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8877303);