#8806799

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

4394 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=ghi,e^2=ghi,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, 8713995);