Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
1912 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,27)(16,31)(17,23)(18,25)(19,24)(20,26)(21,29)(22,28)(30,42)(32,43)(33,39)(34,38)(35,41)(36,40)(37,44);(5,6)(7,11)(8,10)(9,12)(13,15,18,36,20,30,19,29)(14,21,24,42,26,40,25,27)(16,32,37,35,28,17,33,34)(22,41,44,43,31,38,39,23);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,28)(14,22,26,31)(15,17,30,32)(18,37,19,33)(21,38,40,41)(23,27,43,42)(24,44,25,39)(29,35,36,34);(7,12)(9,11)(13,17,20,32)(14,23,26,43)(15,28,30,16)(18,35,19,34)(21,39,40,44)(22,42,31,27)(24,38,25,41)(29,33,36,37);(6,10)(9,11)(13,18,20,19)(14,24,26,25)(15,29,30,36)(16,33,28,37)(17,34,32,35)(21,27,40,42)(22,39,31,44)(23,41,43,38)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910674);