#8932761

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

4585 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839957);