#8772772

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

4461 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679968);