#9913309

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

4825 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820505);