#7656919

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

4557 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564115);