#9436354

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

4979 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9343550);