Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,25)(15,22,28,38)(16,29,27,21)(17,23,33,42)(18,24,19,35)(26,37,31,44)(30,39,43,36)(32,40,34,41);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,16,20,28,19,27)(14,21,35,38,25,29,24,22)(17,31,34,43,33,26,32,30)(23,39,40,37,42,36,41,44);(6,10)(7,11)(13,16,18,28,20,27,19,15)(14,22,35,21,25,38,24,29)(17,30,34,31,33,43,32,26)(23,37,40,36,42,44,41,39);(7,11)(9,12)(13,17)(14,23)(15,26)(16,30)(18,34)(19,32)(20,33)(21,36)(22,37)(24,41)(25,42)(27,43)(28,31)(29,39)(35,40)(38,44);(13,18,20,19)(14,24,25,35)(15,16,28,27)(17,32,33,34)(21,22,29,38)(23,40,42,41)(26,43,31,30)(36,44,39,37)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gh,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366977);