Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2011 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11008 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,12,7,30)(3,14,17,47,18,46,31,42)(4,19,22,56,23,38,32,43)(5,24,27,44,29,59,28,11)(9,25,35,54,34,60,36,16)(10,26,40,20,39,58,41,21)(15,33,50,63,52,37,51,64)(45,57,61,48,55,53,62,49);(1,3)(2,9)(4,15)(5,25)(6,17)(7,31)(8,18)(10,37)(11,42)(12,34)(13,35)(14,24)(16,28)(19,55)(20,57)(21,53)(22,50)(23,52)(26,49)(27,54)(29,60)(30,36)(32,51)(33,39)(38,45)(40,64)(41,63)(43,61)(44,47)(46,59)(48,58)(56,62);(1,4)(2,10)(3,15)(5,26)(6,22)(7,32)(8,23)(9,33)(11,43)(12,39)(13,40)(14,45)(16,53)(17,50)(18,52)(19,24)(20,27)(21,28)(25,49)(29,58)(30,41)(31,51)(34,37)(35,63)(36,64)(38,59)(42,62)(44,56)(46,55)(47,61)(48,60)(54,57)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=eh,h^b=hi,d^c=dhi,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279962, #72280165, #72307187, … |
| maximal normal subgroup of | #72279962, #72280165, #72307187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55082);