#9057915

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

4414 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)(9,10,14,12)(11,15,18,16)(13,20,21,17)(19,23,24,22)(25,26)(27,29)(28,32)(30,34)(31,33)(35,37)(36,38)(39,40);(2,6)(3,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,27,30,33)(26,31,34,29)(28,35,38,40)(32,39,36,37);(1,3)(2,4)(5,8)(6,7)(10,12)(11,18)(13,21)(22,23)(25,26,30,34)(27,31,33,29)(28,36,38,32)(35,37,40,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,37)(39,40);(1,2)(3,4)(5,6)(7,8)(9,12,14,10)(11,16,18,15)(13,20,21,17)(19,23,24,22)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,17,12,20)(11,19,18,24)(15,22,16,23)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,40,36,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965111);