#7718400

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

4390 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)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(26,31)(29,33)(32,38)(36,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,26,30,31)(27,32,37,38)(28,29,34,33)(35,36,40,39);(10,14)(11,18)(16,20)(19,24)(26,31)(28,34)(32,38)(35,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,30,37)(26,32,31,38)(28,35,34,40)(29,36,33,39);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28,30,34)(26,33,31,29)(27,35,37,40)(32,39,38,36);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,33)(26,34,31,28)(27,36,37,39)(32,40,38,35)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7625596);