#9174993

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

4430 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,18)(13,16)(14,17)(19,23)(20,24)(21,22)(26,31)(27,33)(29,37)(36,38);(1,2,5,6)(3,7,8,4)(9,11,14,18)(10,15,17,12)(13,19,22,24)(16,23,21,20)(26,31)(28,34)(32,39)(36,38);(1,3)(2,4)(5,8)(6,7)(11,18)(12,15)(19,24)(20,23)(25,26,30,31)(27,34,33,28)(29,32,37,39)(35,38,40,36);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,20)(14,15)(16,19)(17,18)(21,24)(22,23)(25,27,30,33)(26,28,31,34)(29,35,37,40)(32,36,39,38);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,18)(13,21)(14,17)(16,22)(19,20)(23,24)(25,28)(26,27)(29,38)(30,34)(31,33)(32,40)(35,39)(36,37);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,39)(33,40)(34,38)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=gh,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, 9082189);