#7985990

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

4456 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)(11,17)(12,14)(13,16)(15,18)(19,24)(20,22)(21,23)(26,31)(27,32)(33,37)(34,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,26,30,31)(27,28,32,35)(29,37,39,33)(34,38,40,36);(1,3,5,4)(2,7,6,8)(10,12)(11,18)(16,20)(19,21)(26,31)(28,35)(33,37)(36,38);(1,4,5,3)(2,8,6,7)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,27,30,32)(26,28,31,35)(29,34,39,40)(33,36,37,38);(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,28)(26,32)(27,31)(29,38)(30,35)(33,34)(36,39)(37,40);(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,29,30,39)(26,33,31,37)(27,34,32,40)(28,36,35,38)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=g,f^2=i,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, 7893186);