#9224317

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

4527 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9131513);