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 | |
8842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(7,11)(10,12)(14,19)(16,24)(17,27)(20,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(20,23)(21,26)(22,28);(6,9)(10,12)(13,14)(15,23)(16,26)(17,22)(18,19)(20,25)(21,24)(27,28);(13,15,18,25)(14,20,19,23)(16,17,24,27)(21,22,26,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,24)(14,21,19,26)(15,17,25,27)(20,22,23,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,22)(15,24)(16,25)(18,27)(19,28)(20,26)(21,23)Polycyclic presentation:
2^9|d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304554, #72304569, #72304609, … |
| maximal normal subgroup of | #72304554, #72304569, #72304609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615855);