Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3352 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)(5,6,9,7)(8,11,12,10)(13,14,17,19)(15,20,16,18)(21,22)(23,27)(24,26)(25,28);(5,6)(7,9)(8,11)(10,12)(13,15,17,16)(14,18,19,20)(21,22,24,26)(23,25,28,27);(6,7)(10,11)(15,16)(18,20)(23,28)(25,27);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,17,15)(14,18,19,20)(21,23)(22,25)(24,28)(26,27);(5,8)(6,10)(7,11)(9,12)(13,14)(15,20)(16,18)(17,19)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
| maximal subgroup of | #72327316, #72327371, #72327389, … |
| maximal normal subgroup of | #72327316, #72327371, #72327389, … |
| maximal quotient of | #72330438 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4406662);