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 | |
1736 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)(13,14,18,19)(15,20,21,16)(17,22,25,26)(23,27,28,24)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(7,11)(9,12)(14,19)(15,21)(17,25)(24,27)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,19)(17,23)(18,21)(22,27)(24,26)(25,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,20)(14,21,19,15)(17,24,25,27)(22,28,26,23)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fgh,d^2=fhi,e^2=g,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, 1620391);