Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4562 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)(7,10)(8,11)(9,12)(14,18)(15,20)(21,22)(23,27)(24,25)(26,28);(5,7,8,12)(6,9,11,10)(13,14)(15,16)(17,18)(19,20)(22,25)(26,27);(13,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23,24,28)(22,26,25,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,20)(15,18)(17,19)(21,22)(23,27)(24,25)(26,28)Polycyclic presentation:
2^9|b^2=fh,d^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72263186, #72263194, #72263222, … |
| maximal normal subgroup of | #72263186, #72263194, #72263222, … |
| maximal quotient of | #72328650, #72328771, #72330754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 685864);