Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,7)(10,11)(13,14,17,20)(15,16,19,18)(21,22,24,27)(23,26,28,25);(2,4)(5,6,9,7)(8,10,12,11)(13,15)(14,18)(16,20)(17,19)(21,23,24,28)(22,25,27,26);(1,2)(3,4)(21,23)(22,25)(24,28)(26,27);(5,7,9,6)(8,10,12,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,26)(24,28)(25,27);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,20)(17,18)(21,22)(23,25)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72332213, #72332239, #72332271 |
| maximal normal subgroup of | #72332213, #72332239, #72332271 |
| maximal quotient of | #72333029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2565160);