Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3551 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(13,14)(15,24)(16,28)(17,23)(18,32)(19,22)(20,26)(21,29)(25,40)(27,36)(30,38)(31,37)(33,35)(34,42)(39,41)(43,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,33)(19,27)(21,34)(22,35)(23,36)(24,32)(28,44)(29,41)(30,42)(31,43)(37,39)(38,40);(2,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,31)(18,30)(19,29)(20,34)(22,28)(23,38)(24,39)(26,43)(27,41)(32,37)(33,42)(35,44)(36,40);(5,7,8,11)(6,10,9,12)(14,22)(16,29)(20,35)(23,32)(24,36)(25,41)(30,31)(42,43);(6,9)(10,12)(13,17,19,18)(14,23,22,32)(15,26,27,33)(16,30,29,31)(20,36,35,24)(21,37,28,38)(25,42,41,43)(34,39,44,40)Polycyclic presentation:
2^9|a^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424095);