#9206773

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

4577 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gh,e^2=i,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, 9113969);