Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
2194 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)(3,4)(7,11)(10,12)(13,14,18,24,19,25,17,23)(15,26,28,37,29,21,34,20)(16,22,32,40,33,41,35,42)(27,39,30,44,31,38,43,36);(6,9)(7,11)(13,15)(14,20)(16,30)(17,28)(18,34)(19,29)(21,24)(22,39)(23,26)(25,37)(27,32)(31,35)(33,43)(36,40)(38,41)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,31)(17,34)(18,28)(19,29)(20,24)(22,39)(23,37)(25,26)(27,33)(30,35)(32,43)(36,42)(38,41)(40,44);(6,9)(10,12)(13,16,18,32,19,33,17,35)(14,22,24,40,25,41,23,42)(15,27,28,30,29,31,34,43)(20,36,26,39,37,44,21,38);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,28,29,34)(16,32,33,35)(20,26,37,21)(22,40,41,42)(27,43,31,30)(36,38,44,39)Polycyclic presentation:
2^9|a^2=f,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053324);