Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27)(29,30,34,35)(31,38,36,32)(33,41,44,37)(39,40,42,43);(2,4)(5,6,9,7)(8,10,12,11)(14,18)(15,22)(20,24)(23,25)(30,35)(31,36)(37,41)(39,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,24)(17,18)(20,26)(21,22)(23,28)(25,27)(29,31,34,36)(30,32,35,38)(33,42,44,39)(37,43,41,40);(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,25)(17,22)(18,21)(20,28)(23,26)(24,27)(29,32)(30,36)(31,35)(33,43)(34,38)(37,39)(40,44)(41,42);(13,16,17,26)(14,20,18,24)(15,23,22,25)(19,27,21,28)(29,33)(30,37)(31,39)(32,40)(34,44)(35,41)(36,42)(38,43)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=fgi,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3486968);