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 | |
7989 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)(5,6,9,7)(8,11,12,10)(15,21)(19,25)(22,27)(26,28);(1,2)(3,4)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27);(6,7)(10,11)(14,18)(15,21)(20,23)(22,27);(5,7,9,6)(8,11,12,10)(13,15,17,21)(14,19,18,25)(16,22,24,27)(20,26,23,28);(5,8,9,12)(6,10,7,11)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326660, #72333436 |
| maximal normal subgroup of | #72326660, #72333436 |
| maximal quotient of | #72326670, #72326880, #72327263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7648714);