Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3842 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)(5,6,8,10)(7,11,12,9)(13,14)(15,17)(16,18)(19,20)(21,22,24,26)(23,25,28,27);(5,6)(7,11)(8,10)(9,12)(13,14,17,15)(16,18,20,19)(21,22)(23,27)(24,26)(25,28);(7,12)(9,11)(14,15)(16,20)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,14)(16,19,20,18)(21,22)(23,25)(24,26)(27,28);(5,7,8,12)(6,9,10,11)(13,16,17,20)(14,18,15,19)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=gh,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72300470, #72300488, #72300555, … |
| maximal normal subgroup of | #72300470, #72300488, #72300555, … |
| maximal quotient of | #72330785 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5388236);