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 | |
7649 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)(7,10)(8,12)(9,11)(13,14)(15,22)(16,19)(17,25)(18,20)(21,26)(23,27)(24,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,16)(14,19,20,22)(17,23,26,24)(21,27,25,28);(6,11)(10,12)(14,20)(15,16)(21,25)(23,24);(5,6)(7,10)(8,12)(9,11)(13,16,18,15)(14,19,20,22)(17,24,26,23)(21,27,25,28);(5,8,9,7)(6,12,11,10)(13,14)(15,22)(16,19)(17,21)(18,20)(23,28)(24,27)(25,26);(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,25)(22,28)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=hi,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72323293, C2×#7678839 |
| maximal normal subgroup of | #72323293, C2×#7678839 |
| maximal quotient of | #72323329, #72323340, #72323403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7586035);