Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2908 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)(13,14)(15,22)(16,17)(18,21)(19,27)(20,25)(23,28)(24,26);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,26)(17,28)(18,25)(19,24)(21,22)(23,27);(6,10)(7,11)(13,16,17,14,18,27,19,21)(15,23,28,22,25,26,24,20);(7,11)(9,12)(13,17,18,19)(14,16,21,27)(15,24,25,28)(20,23,22,26);(15,25)(20,22)(23,26)(24,28)Polycyclic presentation:
2^9|c^2=gh,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216148 |
| maximal normal subgroup of | #72216148 |
| maximal quotient of | #72216022 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366550);