Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(14,19)(16,17)(20,26)(21,25)(22,24)(27,28);(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,19)(17,25)(18,21)(22,27)(23,28)(24,26);(1,2,3,4)(6,9)(7,11)(13,15)(14,20)(16,24)(17,22)(18,23)(19,26)(21,28)(25,27);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,19,21,25)(15,22,23,24)(20,27,28,26);(6,9)(7,11)(15,23)(20,28)(22,24)(26,27)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72307388, #72307677, #72307855 |
| maximal normal subgroup of | #72307388, #72307677, #72307855 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513458);