Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11022 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
957 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)(3,14)(4,19,23,9)(5,25)(6,31)(7,11)(8,16)(10,29)(12,43)(13,28)(15,49,51,63)(17,35)(18,44)(20,58,50,64)(21,42,22,45)(24,40,33,48)(26,47)(27,39,59,57)(30,46)(32,41)(34,60,62,53)(36,61)(37,54)(38,52,56,55);(1,3)(2,8)(4,20)(5,26)(6,32)(7,17)(9,38)(10,41)(11,37)(12,44)(13,47)(14,25)(15,22)(16,29)(18,28)(19,56)(21,51)(23,50)(24,60)(27,55)(30,61)(31,36)(33,53)(34,39)(35,46)(40,49)(42,58)(43,54)(45,64)(48,63)(52,59)(57,62);(1,4)(2,9)(3,15)(5,27)(6,33)(7,23)(8,34)(10,42)(11,19)(12,45)(13,48)(14,49)(16,53)(17,51)(18,55)(20,26)(21,43)(22,29)(24,28)(25,57)(30,59)(31,40)(32,52)(35,63)(36,64)(37,62)(38,44)(39,46)(41,56)(47,58)(50,61)(54,60)Polycyclic presentation:
2^9|a^2=g,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=dhi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307198, #72307228, #72307275, … |
| maximal normal subgroup of | #72307198, #72307228, #72307275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55140);