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 | |
8644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,8,10)(7,9,12,11)(13,14,18,20)(15,19,23,28)(16,21,25,27)(17,22,24,26);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,23)(14,19,20,28)(16,24,25,17)(21,26,27,22);(5,6)(7,11)(8,10)(9,12)(14,20)(15,23)(16,25)(22,26);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,24)(17,23)(18,25)(19,26)(20,27)(22,28);(13,17,18,24)(14,22,20,26)(15,25,23,16)(19,27,28,21);(13,14)(15,19)(16,21)(17,22)(18,20)(23,28)(24,26)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72302317, #72302972, #72302988, … |
| maximal normal subgroup of | #72302317, #72302972, #72302988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589920);