#7986220

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,e^2=h,f^2=gi,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, 7893416);