#7901237

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

4321 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=gi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7808433);