#9206444

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

4300 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)(4,7,8,6)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40);(2,3)(6,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,27,31,34)(26,30,35,36)(28,37,33,40)(29,38,32,39);(1,3,5,2)(4,7,8,6)(10,12)(16,18)(17,20)(23,24)(25,28,31,33)(26,32,35,29)(27,37,34,40)(30,39,36,38);(1,4)(2,6)(3,7)(5,8)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,28)(26,32)(27,37)(29,35)(30,39)(31,33)(34,40)(36,38);(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,32)(26,33,35,28)(27,38,34,39)(30,40,36,37);(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,30)(26,34)(27,35)(28,39)(29,37)(31,36)(32,40)(33,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113640);