Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1598 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)(5,6,9,10)(7,8,12,11)(13,14,18,19)(15,16,22,20)(17,21,26,27)(23,24,25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(5,7)(6,8)(9,12)(10,11)(14,19)(15,22)(21,27)(23,25)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(6,10)(7,12)(13,15,18,22)(14,20,19,16)(17,25,26,23)(21,24,27,28)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,20)(14,15,19,22)(17,24,26,28)(21,23,27,25)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,8,9,11)(6,7,10,12)(13,17,18,26)(14,21,19,27)(15,23,22,25)(16,24,20,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5383079);