Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,33,18,24,17,23)(15,20,34,43,28,38,27,37)(16,29,30,21,32,22,31,39)(25,36,41,35,26,40,42,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,34)(21,35)(22,40)(23,37)(24,38)(26,32)(29,36)(30,41)(31,42)(33,43)(39,44);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,35)(22,23)(24,39)(26,28)(27,41)(29,33)(34,42)(36,43)(37,40)(38,44);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,26)(17,31)(18,32)(19,30)(20,36)(21,23)(24,29)(25,28)(27,41)(33,39)(34,42)(35,43)(37,44)(38,40);(6,10)(9,11)(13,17,18,19)(14,23,24,33)(15,27,28,34)(16,30,32,31)(20,37,38,43)(21,22,39,29)(25,41,26,42)(35,40,44,36)Polycyclic presentation:
2^9|a^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913796);