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 | |
7376 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)(6,9)(10,11)(13,14)(15,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12);(5,6,8,9)(7,10,12,11)(14,19)(15,20)(21,26)(22,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,16,19,24)(17,22,25,27)(21,23,26,28);(5,6)(7,11)(8,9)(10,12)(13,16)(14,20)(15,19)(17,23)(18,24)(21,27)(22,26)(25,28);(5,7)(6,10)(8,12)(9,11)(13,17,18,25)(14,21,19,26)(15,22,20,27)(16,23,24,28)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=hi,f^2=hi,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 | #72307519, #72307686, #72308489, … |
| maximal normal subgroup of | #72307519, #72307686, #72308489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894509);