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 | |
7864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(13,14)(15,17)(16,23)(18,19)(20,24)(21,26)(22,28)(25,27);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,18,21)(14,17,19,26)(16,22,24,27)(20,25,23,28);(6,9)(10,11)(14,19)(15,21)(16,24)(25,28);(13,14)(15,17)(16,20)(18,19)(21,26)(22,25)(23,24)(27,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,23)(21,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,15)(16,25)(18,26)(19,21)(20,22)(23,27)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72327652, #72331611 |
| maximal normal subgroup of | #72327652, #72331611 |
| maximal quotient of | #72327735, #72328971, #72329491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7629048);