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 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1991 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)(7,11)(10,12)(13,14,17,23,18,22,29,33)(15,20,26,35,25,37,32,19)(16,21,31,39,30,40,41,44)(24,36,27,42,28,38,43,34);(6,9)(7,11)(13,15,18,25)(14,19,22,35)(16,27,30,43)(17,32,29,26)(20,23,37,33)(21,36,40,38)(24,41,28,31)(34,44,42,39);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,28)(17,26)(18,25)(19,33)(21,38)(22,37)(23,35)(24,30)(27,41)(29,32)(31,43)(34,39)(36,40)(42,44);(6,9)(10,12)(13,16,29,41,18,30,17,31)(14,21,33,44,22,40,23,39)(15,24,32,43,25,28,26,27)(19,34,37,38,35,42,20,36);(5,7,8,11)(6,10,9,12)(14,22)(15,25)(16,30)(23,33)(26,32)(31,41)(34,42)(36,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fi,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=dh,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, 7063512);