#8756140

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

4432 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)(4,5)(6,8)(9,10)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(26,28)(27,30)(29,37)(39,40);(2,5)(6,7)(10,12)(13,21)(15,17)(19,24)(25,26,31,28)(27,32,30,35)(29,33,37,36)(34,39,38,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,24)(22,23)(25,27,31,30)(26,32,28,35)(29,34,37,38)(33,39,36,40);(1,3,4,8)(2,6,5,7)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,28,31,26)(27,32,30,35)(29,36,37,33)(34,39,38,40);(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,27)(26,32,28,35)(29,38,37,34)(33,39,36,40)

Polycyclic presentation:

2^9|b^2=h,c^2=h,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, 8663336);