Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2554 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)(7,9)(8,10)(11,12)(13,14)(15,21)(16,26)(17,19)(18,24)(20,22)(23,28)(25,27);(5,7)(6,9)(8,11)(10,12)(13,15,22,28,17,25,16,24)(14,18,26,27,19,23,20,21);(1,2,3,4)(7,11)(9,12)(13,14,17,19)(15,23,25,18)(16,20,22,26)(21,24,27,28);(7,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,26)(23,25)(27,28);(6,10)(9,12)(14,19)(18,23)(20,26)(21,27)Polycyclic presentation:
2^9|b^2=fi,c^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380473);