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 | |
7101 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)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15,18,16)(14,19,22,20)(17,23,25,24)(21,26,28,27);(7,11)(8,12)(13,15,18,16)(14,20,22,19)(17,23,25,24)(21,27,28,26);(6,10)(8,12)(13,16,18,15)(14,20,22,19)(17,23,25,24)(21,26,28,27);(5,8)(6,7)(9,12)(10,11)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28);(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^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, #72282706, #72282718, … |
| maximal normal subgroup of | #72282562, #72282706, #72282718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7970344);