Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,11)(7,10,12,8)(13,14,17,21)(15,19,23,18)(16,24,25,20)(22,27,28,26);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,20)(17,23)(19,21)(22,26)(24,25)(27,28);(6,11)(7,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(5,8,9,10)(6,12,11,7)(13,16,17,25)(14,20,21,24)(15,22,23,28)(18,27,19,26);(1,3,4,2)(14,21)(15,23)(20,24)(22,28)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72263913, #72263918, #72264061 |
| maximal normal subgroup of | #72263913, #72263918, #72264061 |
| maximal quotient of | #72264257 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7464766);