#8756056

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

4657 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8663252);