Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| 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 | |
5666 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)(7,11)(8,9)(10,12)(13,14,17,20)(15,19,16,18)(21,22,25,27)(23,24,28,26);(6,9)(10,11)(13,15,17,16)(14,18,20,19)(21,23,25,28)(22,26,27,24);(6,9)(10,11)(15,16)(18,19)(22,27)(23,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,17,15)(14,19,20,18)(21,23)(22,24)(25,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,24)(22,23)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232308, #72232330, #72232360, … |
| maximal normal subgroup of | #72232308, #72232330, #72232360, … |
| maximal quotient of | #72328841, #72329026, #72329952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 825537);