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 | |
8386 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)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(2,4)(5,6)(7,11)(8,9)(10,12)(14,18)(16,25)(19,21)(22,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|d^2=g,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 | #72249482, #72249486, #72249502, … |
| maximal normal subgroup of | #72249482, #72249486, #72249502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589131);