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 | |
7746 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)(15,20)(16,19)(17,22)(18,21)(23,26)(24,27)(25,28);(1,2,3,4)(13,15,19,22)(14,20,16,17)(18,23,27,28)(21,26,24,25);(6,9)(10,11)(14,16)(15,22)(21,24)(23,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,14)(15,17,22,20)(18,24,27,21)(23,25,28,26);(5,7)(6,10)(8,12)(9,11)(13,17)(14,15)(16,22)(18,26)(19,20)(21,28)(23,24)(25,27);(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,26)(22,28)Polycyclic presentation:
2^9|b^2=gi,d^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72326660, #72326930 |
| maximal normal subgroup of | #72326660, #72326930 |
| maximal quotient of | #72326754, #72326819, #72327002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7841281);