Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 7340032 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
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 | |
10332 conjugacy classes of subgroups; only the 546 normal subgroups are shown.
This group has 10332 conjugacy classes of subgroups — too many to draw. Only its 546 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(14,17)(15,19)(22,25)(26,28);(1,2)(3,4)(6,9)(10,12)(15,19)(18,20)(22,25)(23,27);(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
| maximal subgroup of | #72327697, #72327710, #72327730, … |
| maximal normal subgroup of | #72327697, #72327710, #72327730, … |
| maximal quotient of | #72330154 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038831);