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 | |
7097 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)(7,8)(9,10)(11,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15,18,24)(14,19,20,26)(16,17,23,25)(21,22,27,28);(7,11)(8,12)(14,20)(19,26)(21,27)(22,28);(6,10)(8,12)(13,16)(14,21)(15,17)(18,23)(19,22)(20,27)(24,25)(26,28);(5,8)(6,7)(9,12)(10,11)(13,17)(14,22)(15,23)(16,24)(18,25)(19,27)(20,28)(21,26);(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72282562, #72282607, #72282617, … |
| maximal normal subgroup of | #72282562, #72282607, #72282617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7970132);