Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,7)(8,10,12,11)(13,14)(15,16)(17,18)(19,20)(22,26)(23,29)(24,32)(34,36);(2,4)(6,7)(10,11)(14,18)(15,20)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(5,8,9,12)(6,10,7,11)(13,16,17,19)(14,15,18,20)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=fg,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863663);