#9007398

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

4657 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=i,f^2=h,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, 8914594);