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 | |
2191 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)(13,14,19,23)(15,20,26,36)(16,27,25,41)(17,30,32,44)(18,34,28,22)(21,37,40,33)(24,38,42,43)(29,39,31,35);(1,2)(3,4)(6,10)(11,12)(13,15,19,26)(14,20,23,36)(16,28,25,18)(17,31,32,29)(21,38,40,43)(22,41,34,27)(24,37,42,33)(30,35,44,39);(5,7,9,8)(6,11,10,12)(13,16,28,26,19,25,18,15)(14,20,34,41,23,36,22,27)(17,31,37,24,32,29,33,42)(21,39,44,38,40,35,30,43);(2,4)(5,8,9,7)(6,12,10,11)(13,17,28,37,19,32,18,33)(14,21,34,44,23,40,22,30)(15,24,16,29,26,42,25,31)(20,35,41,43,36,39,27,38);(6,10)(11,12)(17,32)(21,40)(24,42)(29,31)(30,44)(33,37)(35,39)(38,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fgi,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,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469473);