#9167693

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

4604 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26,30,32)(27,31,34,28)(29,37,39,33)(35,36,40,38);(2,7)(3,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,30,34)(26,31,32,28)(29,35,39,40)(33,38,37,36);(1,3,6,8)(2,4,7,5)(11,18)(14,15)(19,24)(22,23)(26,32)(27,34)(33,37)(35,40);(1,4,6,5)(2,3,7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,26)(27,31)(28,34)(29,33)(30,32)(35,38)(36,40)(37,39);(1,2)(3,4)(5,8)(6,7)(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,28,30,31)(26,27,32,34)(29,38,39,36)(33,40,37,35);(1,5,6,4)(2,8,7,3)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,37)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=gh,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, 9074889);