#8970205

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

4300 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)(11,17)(12,22)(13,15)(14,19)(16,20)(18,23)(21,24)(25,26,31,36)(27,35,30,32)(28,29,34,33)(37,40,39,38);(1,2)(3,4)(5,7)(6,8)(9,11,16,13)(10,17,20,15)(12,14,23,21)(18,24,22,19)(25,27,31,30)(26,32,36,35)(28,39,34,37)(29,40,33,38);(1,3)(2,6)(4,7)(5,8)(9,12,16,23)(10,18,20,22)(11,21,13,14)(15,24,17,19)(25,28,31,34)(26,33,36,29)(27,37,30,39)(32,40,35,38);(3,8)(4,6)(9,13,16,11)(10,15,20,17)(12,21,23,14)(18,19,22,24)(25,27,31,30)(26,32,36,35)(28,37,34,39)(29,38,33,40);(1,4)(2,3)(5,6)(7,8)(9,14,16,21)(10,19,20,24)(11,12,13,23)(15,18,17,22)(25,29,31,33)(26,34,36,28)(27,38,30,40)(32,39,35,37);(1,2)(3,4)(5,7)(6,8)(9,15)(10,11)(12,19)(13,20)(14,18)(16,17)(21,22)(23,24)(25,30,31,27)(26,35,36,32)(28,39,34,37)(29,40,33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,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, 8877401);