#9781112

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

4545 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9688308);