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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,19)(16,17)(20,28)(21,23)(22,27)(24,26);(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,19)(17,27)(18,22)(20,25)(21,26)(24,28);(2,4)(5,7,8,12)(6,9,10,11)(13,15,18,25)(14,20,22,23)(16,26,17,24)(19,28,27,21);(7,12)(9,11)(13,15)(14,21)(16,24)(17,26)(18,25)(19,23)(20,27)(22,28);(6,10)(9,11)(13,16,18,17)(14,19,22,27)(15,24,25,26)(20,21,23,28)Polycyclic presentation:
2^9|c^2=h,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=ehi,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283309, #72283416, #72285000, … |
| maximal normal subgroup of | #72283309, #72283416, #72285000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961944);