#9105271

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

4509 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)(11,21)(12,18)(13,15)(14,19)(16,20)(17,22)(23,24)(25,26,31,29)(27,28,35,32)(30,37,39,33)(34,38,40,36);(1,2,5,3)(4,6,8,7)(9,11,16,22)(10,17,20,21)(12,19,13,23)(14,18,24,15)(26,29)(28,32)(33,37)(36,38);(2,3)(6,7)(9,12,16,13)(10,18,20,15)(11,19,22,23)(14,17,24,21)(25,27,31,35)(26,32,29,28)(30,34,39,40)(33,38,37,36);(2,3)(4,8)(9,13,16,12)(10,18,20,15)(11,19,22,23)(14,21,24,17)(25,28)(26,27)(29,35)(30,38)(31,32)(33,40)(34,37)(36,39);(1,3,5,2)(4,7,8,6)(9,14)(10,19)(11,18)(12,21)(13,17)(15,22)(16,24)(20,23)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,38,40,36);(1,4)(2,6)(3,7)(5,8)(9,15,16,18)(10,13,20,12)(11,14,22,24)(17,23,21,19)(25,30,31,39)(26,33,29,37)(27,34,35,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^2=hi,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, 9012467);