#8097768

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

4815 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)(3,4)(5,6)(7,8)(12,20)(15,22)(18,24)(21,23)(25,26)(27,29)(28,37)(30,33)(31,34)(32,36)(35,40)(38,39);(2,6)(3,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,22)(21,24)(25,27)(26,32)(28,38)(29,34)(30,35)(31,36)(33,39)(37,40);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,31,30)(26,33,34,37)(27,35,36,38)(29,39,32,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,29)(26,27)(28,39)(30,40)(31,32)(33,35)(34,36)(37,38);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,30,31,28)(26,33,34,37)(27,35,36,38)(29,40,32,39);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,22)(23,24)(25,27)(26,29)(28,35)(30,38)(31,36)(32,34)(33,39)(37,40)

Polycyclic presentation:

2^9|c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8004964);