Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,24)(17,20)(18,23)(21,26)(22,28)(25,27);(5,7)(6,9)(8,11)(10,12)(13,15,17,22)(14,18,20,27)(16,25,26,23)(19,21,28,24);(2,4)(7,11)(9,12)(13,15)(14,19)(16,23)(17,22)(18,24)(20,28)(21,27)(25,26);(7,11)(9,12)(15,22)(18,27)(19,28)(23,25);(6,10)(9,12)(14,20)(18,27)(19,28)(21,24)Polycyclic presentation:
2^9|b^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72315567, #72315570, #72315587, … |
| maximal normal subgroup of | #72315567, #72315570, #72315587, … |
| maximal quotient of | #72316127, #72316232, #72316390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377154);