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 | |
2172 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)(13,14,19,25,20,26,18,24)(15,22,28,37,16,29,31,21)(17,23,33,40,34,41,35,42)(27,38,32,44,30,39,43,36);(6,9)(7,11)(13,15,20,16)(14,21,26,37)(17,32,34,43)(18,28,19,31)(22,25,29,24)(23,38,41,39)(27,35,30,33)(36,42,44,40);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,15)(14,22,26,29)(17,27,34,30)(18,28,19,31)(21,25,37,24)(23,39,41,38)(32,35,43,33)(36,42,44,40);(6,9)(10,12)(13,17,18,35,20,34,19,33)(14,23,24,42,26,41,25,40)(15,27,31,43,16,30,28,32)(21,36,29,39,37,44,22,38);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,26,25)(15,28,16,31)(17,33,34,35)(21,22,37,29)(23,40,41,42)(27,43,30,32)(36,39,44,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,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=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, 7053338);