#9184121

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

4769 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,f^2=g,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, 9091317);