#9067525

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

4370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,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,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8974721);