Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,9)(7,11)(14,15)(17,25)(19,21)(23,28);(1,2,3,4)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)(13,14,18,15)(16,19,24,21)(17,20,25,22)(23,26,28,27);(5,7,8,11)(6,10,9,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72256602, #72256689, #72256691 |
| maximal normal subgroup of | #72256602, #72256689, #72256691 |
| maximal quotient of | #72256782, #72263480 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7552592);