#9078801

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

4310 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)(11,14)(12,15)(13,17)(16,19)(18,23)(20,24)(21,22)(25,26,31,36)(27,34,29,32)(28,30,35,33)(37,40,39,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,20)(13,19)(15,24)(16,17)(18,22)(21,23)(25,27,31,29)(26,32,36,34)(28,39,35,37)(30,40,33,38);(2,5)(6,8)(9,10,13,17)(11,16,19,14)(12,15,22,21)(18,24,20,23)(25,28,31,35)(26,33,36,30)(27,37,29,39)(32,40,34,38);(2,5)(3,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,24)(17,21)(19,20)(25,29,31,27)(26,34,36,32)(28,39,35,37)(30,40,33,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,21)(13,17)(14,19)(15,22)(18,23)(20,24)(25,30,31,33)(26,35,36,28)(27,38,29,40)(32,39,34,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,19)(14,17)(15,23)(18,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8985997);