Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,20)(16,22)(17,21)(18,19)(23,27)(24,28)(25,26)(29,30,34,37)(31,33,39,35)(32,40,43,36)(38,44,41,42);(7,11)(9,12)(14,19)(15,22)(17,25)(24,27)(29,31,34,39)(30,35,37,33)(32,41,43,38)(36,42,40,44);(5,7)(6,9)(8,11)(10,12)(13,15,18,22)(14,20,19,16)(17,23,25,28)(21,27,26,24)(29,31)(30,35)(32,41)(33,37)(34,39)(36,42)(38,43)(40,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,20)(14,15,19,22)(17,24,25,27)(21,23,26,28)(29,32,34,43)(30,36,37,40)(31,38,39,41)(33,42,35,44);(13,17,18,25)(14,21,19,26)(15,23,22,28)(16,24,20,27)(29,33,34,35)(30,31,37,39)(32,42,43,44)(36,38,40,41)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1652999);