Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4428 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)(6,9)(7,11)(13,14)(15,19)(16,18)(17,20)(21,22)(23,28)(24,26)(25,27);(5,6)(7,12)(8,9)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23,25,24)(22,26,27,28);(7,11)(10,12)(14,18)(17,19)(21,22,25,27)(23,28,24,26);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,28)(25,27);(5,6)(7,12)(8,9)(10,11)(13,15)(14,19)(16,20)(17,18)(21,24,25,23)(22,26,27,28)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72231820, #72232640, #72232653, … |
| maximal normal subgroup of | #72231820, #72232640, #72232653, … |
| maximal quotient of | #72237912, #72278125, #72285028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2672217);