Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
8587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,9,12,11)(13,14,19,21)(15,18,24,20)(16,22,25,26)(17,23,27,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,17)(18,21)(19,24)(22,28)(23,26)(25,27);(2,4)(5,6)(7,11)(8,10)(9,12)(14,21)(15,24)(16,25)(23,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,17)(18,23)(19,25)(20,28)(21,26)(24,27);(13,17)(14,23)(15,16)(18,22)(19,27)(20,26)(21,28)(24,25);(13,18)(14,24)(15,21)(16,23)(17,26)(19,20)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=g,d^2=gi,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72256672, #72256679, #72256684, … |
| maximal normal subgroup of | #72256672, #72256679, #72256684, … |
| maximal quotient of | #72297884, #72297968, #72298087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7594058);