Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2690 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)(5,6)(7,11)(8,10)(9,12)(14,19)(20,27)(21,22)(23,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,23)(16,22)(17,19)(18,21)(20,24)(25,28)(26,27);(1,2,3,4)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(6,10)(9,11)(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,23,27,28);(7,12)(9,11)(14,22)(16,18)(19,21)(20,28)(23,27)(24,26)Polycyclic presentation:
2^9|c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281541, #72282296, #72282353, … |
| maximal normal subgroup of | #72281541, #72282296, #72282353, … |
| maximal quotient of | #72315893 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589962);