Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3614 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)(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,26)(25,28);(5,6)(7,10)(8,9)(11,12)(14,19)(15,17)(20,22)(21,26)(23,24)(27,28);(2,4)(6,9)(7,11)(13,15,18,17)(14,20,22,19)(16,23,25,24)(21,27,28,26);(5,7,8,11)(6,10,9,12)(13,16,18,25)(14,21,22,28)(15,23,17,24)(19,26,20,27);(6,9)(10,12)(13,15,18,17)(14,20,22,19)(16,24,25,23)(21,26,28,27)Polycyclic presentation:
2^9|c^2=i,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257074, #72257111, #72257170, … |
| maximal normal subgroup of | #72257074, #72257111, #72257170, … |
| maximal quotient of | #72265656 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419892);