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 | |
7691 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)(14,19)(15,20)(21,25)(23,26);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,22)(16,20)(17,25)(18,19)(21,27)(23,24)(26,28);(6,9)(10,12)(13,15,18,20)(14,16,19,22)(17,26,27,23)(21,28,25,24);(7,11)(10,12)(13,16,18,22)(14,20,19,15)(17,24,27,28)(21,26,25,23);(5,7)(6,10)(8,11)(9,12)(13,14)(15,16)(17,21)(18,19)(20,22)(23,24)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,17)(14,21)(15,23)(16,24)(18,27)(19,25)(20,26)(22,28)Polycyclic presentation:
2^9|b^2=hi,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72326930 |
| maximal normal subgroup of | #72326930 |
| maximal quotient of | #72326984, #72328391, #72329600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7727109);