#7973814

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7881010);