Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,28,36)(16,21,31,39)(17,22,18,23)(25,33,41,43)(26,34,27,35)(29,37,42,44)(30,38,32,40);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,29)(17,27)(18,26)(19,28)(21,37)(22,35)(23,34)(24,36)(25,30)(31,42)(32,41)(33,38)(39,44)(40,43);(1,2,3,4)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,38)(23,40)(24,39)(26,29)(27,42)(28,41)(34,37)(35,44)(36,43);(6,10)(9,11)(13,17,19,18)(14,22,24,23)(15,26,28,27)(16,30,31,32)(20,34,36,35)(21,38,39,40)(25,29,41,42)(33,37,43,44);(7,12)(9,11)(15,26)(17,18)(20,34)(22,23)(25,29)(27,28)(30,32)(33,37)(35,36)(38,40)(41,42)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590059);