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 |
| prime radical | #36127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,8,9)(7,10,12,11)(16,24)(21,25)(23,26)(27,28);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,17)(15,22,18,20)(16,21,24,25)(23,28,26,27);(6,9)(10,11)(13,15,19,18)(14,20,17,22)(16,23,24,26)(21,27,25,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,24)(14,21,17,25)(15,23,18,26)(20,27,22,28);(5,6)(7,11)(8,9)(10,12)(13,17,19,14)(15,20,18,22)(16,25,24,21)(23,27,26,28);(5,7,8,12)(6,10,9,11)(13,18,19,15)(14,22,17,20)(16,26,24,23)(21,28,25,27)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=hi,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72312245, #72312255, #72312575, … |
| maximal normal subgroup of | #72312245, #72312255, #72312575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7896903);