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 | |
7753 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)(5,6,9,8)(7,10,11,12)(14,19)(15,21)(22,27)(23,25);(1,2)(3,4)(13,14)(15,20)(16,21)(17,22)(18,19)(23,28)(24,25)(26,27);(6,8)(10,12)(14,19)(16,20)(22,27)(24,28);(7,11)(10,12)(13,15)(14,20)(16,19)(17,25)(18,21)(22,24)(23,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,16,18,20)(14,21,19,15)(17,24,26,28)(22,25,27,23);(5,8,9,6)(7,12,11,10)(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,28)(21,25)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^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 | #72263434 |
| maximal normal subgroup of | #72263434 |
| maximal quotient of | #72263465 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7974077);