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 | |
1746 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)(7,11)(9,12)(13,14,18,19)(15,20,21,16)(17,22,27,28)(23,26,25,24)(29,30)(31,37)(32,38)(33,41)(34,35)(36,44)(39,40)(42,43);(5,6,8,10)(7,9,11,12)(14,19)(16,20)(22,28)(24,26)(30,35)(31,38)(36,41)(39,42);(5,6)(7,9)(8,10)(11,12)(13,15,18,21)(14,20,19,16)(17,25,27,23)(22,24,28,26)(29,31)(30,32)(33,42)(34,38)(35,37)(36,43)(39,44)(40,41);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,16,18,20)(14,21,19,15)(17,26,27,24)(22,23,28,25)(29,32)(30,31)(33,43)(34,37)(35,38)(36,42)(39,41)(40,44);(13,17,18,27)(14,22,19,28)(15,23,21,25)(16,24,20,26)(29,33)(30,36)(31,39)(32,40)(34,44)(35,41)(37,43)(38,42)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fg,d^2=hi,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, 3376990);