#7883463

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

4578 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790659);