Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#560 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14)(15,22)(16,25)(17,23)(18,19)(20,26)(21,27)(24,28);(6,9)(10,11)(14,19)(16,27)(20,22)(23,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,28)(23,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,21)(15,24)(17,22)(18,27)(19,25)(20,28)(23,26);(1,2,3,4)(13,15)(14,22)(16,24)(17,21)(18,26)(19,20)(23,27)(25,28);(13,17)(14,23)(15,25)(16,22)(18,28)(19,24)(20,27)(21,26)Polycyclic presentation:
2^9|e^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
| maximal subgroup of | #72323293, #72327640, C2×#7627401 |
| maximal normal subgroup of | #72323293, #72327640, C2×#7627401 |
| maximal quotient of | #72323381, #72323421, #72327884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534597);