#8944511

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

4670 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8851707);