#9105285

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

4315 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 9012481);