Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
3020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,14,18,17)(15,20,16,19)(21,22,25,24)(23,26,27,28);(2,4)(6,10)(8,11)(13,15,18,16)(14,19,17,20)(21,22)(23,26)(24,25)(27,28);(1,2,3,4)(6,10)(8,11)(14,17)(19,20)(22,24)(26,28);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,15)(14,20,17,19)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,8,9,11)(6,12,10,7)(13,17,18,14)(15,19,16,20)(21,24,25,22)(23,26,27,28)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=fg,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72300258, #72300281, #72300283, … |
| maximal normal subgroup of | #72300258, #72300281, #72300283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3886901);