#10041405

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

4461 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)(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,35)(27,33,38,29)(28,32,30,34)(36,39,37,40);(1,3,5,8)(2,4,6,7)(12,19)(16,22)(18,23)(20,24)(25,27,31,38)(26,29,35,33)(28,36,30,37)(32,40,34,39);(2,6)(3,8)(10,15)(11,17)(16,22)(18,23)(25,28,31,30)(26,32,35,34)(27,36,38,37)(29,40,33,39);(1,3)(2,7)(4,6)(5,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,27)(26,33)(28,36)(29,35)(30,37)(31,38)(32,39)(34,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,33)(26,27,35,38)(28,39,30,40)(32,37,34,36);(1,4)(2,8)(3,6)(5,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30,31,28)(26,34,35,32)(27,37,38,36)(29,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=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, 9948601);