Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,12)(9,10)(13,14,20,26)(15,21,29,22)(16,30,28,38)(17,32,36,41)(18,24,19,25)(23,35,42,34)(27,43,44,40)(31,37,33,39);(1,2)(3,4)(6,10)(11,12)(13,15,20,29)(14,21,26,22)(16,18,28,19)(17,33,36,31)(23,40,42,43)(24,38,25,30)(27,34,44,35)(32,39,41,37);(5,7,9,8)(6,11,10,12)(13,16,19,15,20,28,18,29)(14,22,24,38,26,21,25,30)(17,31,35,27,36,33,34,44)(23,39,41,43,42,37,32,40);(2,4)(5,8,9,7)(6,12,10,11)(13,17,19,35,20,36,18,34)(14,23,24,41,26,42,25,32)(15,27,28,33,29,44,16,31)(21,37,30,40,22,39,38,43);(6,10)(11,12)(13,18,20,19)(14,24,26,25)(15,28,29,16)(17,34,36,35)(21,38,22,30)(23,41,42,32)(27,33,44,31)(37,43,39,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469378);