#7709984

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

4449 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)(13,21)(15,17)(18,22)(19,23)(20,24)(28,35)(32,37)(33,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,24,21,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,3,5,4)(2,7,6,8)(10,17)(11,12)(13,22)(23,24)(26,30)(31,36)(32,37)(38,40);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,23,21,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,19)(12,20)(14,23)(15,22)(16,24)(17,21)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,4,5,3)(2,8,6,7)(9,14)(10,11)(12,17)(13,23)(15,16)(18,19)(20,21)(22,24)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617180);