#7719183

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

4387 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)(11,16)(12,14)(13,20)(15,18)(17,21)(19,22)(23,24)(29,37)(32,36)(34,40)(38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3,5,4)(2,7,6,8)(10,12)(11,18)(17,20)(19,24)(26,28)(31,33)(32,36)(38,39);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,12,20)(11,19,18,24)(15,22,16,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gi,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, 7626379);