#9105317

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

4173 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=h,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, 9012513);