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 | |
9148 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)(7,11)(8,9)(10,12)(13,14,18,20)(15,21,22,19)(16,25,27,17)(23,24,26,28);(1,2)(3,4)(13,15)(14,19)(16,23)(17,24)(18,22)(20,21)(25,28)(26,27);(6,9)(10,11)(14,20)(15,22)(17,25)(23,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,26)(17,24)(18,22)(19,20)(23,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,17)(15,23)(18,27)(19,24)(20,25)(21,28)(22,26);(13,17)(14,16)(15,24)(18,25)(19,23)(20,27)(21,26)(22,28)Polycyclic presentation:
2^9|a^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72256668, #72256696, #72256699, … |
| maximal normal subgroup of | #72256668, #72256696, #72256699, … |
| maximal quotient of | #72326938, #72326959, #72327943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7671989);