Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4522 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)(8,12)(9,10)(14,18)(15,16)(21,22)(23,24)(25,27)(26,28);(6,10)(7,12)(13,14,17,18)(15,19,16,20)(21,23,25,28)(22,24,27,26);(5,7)(6,8)(9,12)(10,11)(13,15,17,16)(14,19,18,20)(21,22,25,27)(23,26,28,24);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,27)(25,28);(5,8)(6,7)(9,11)(10,12)(13,16,17,15)(14,20,18,19)(21,24,25,26)(22,23,27,28)Polycyclic presentation:
2^9|b^2=fh,c^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72231339, #72231349, #72231359, … |
| maximal normal subgroup of | #72231339, #72231349, #72231359, … |
| maximal quotient of | #72278057, C2×#850480, #72299653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 757676);