Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| 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:
(1,2,6,14,8,15,32,12)(3,16,21,43,22,47,19,36)(4,23,28,42,29,46,26,10)(5,13,30,11,31,45,7,33)(9,20,35,18,38,55,37,56)(17,50,52,49,54,57,53,39)(24,58,59,64,60,63,51,34)(25,48,27,44,62,40,61,41);(1,3)(2,9)(4,24)(5,20)(6,19)(7,18)(8,22)(10,39)(11,43)(12,35)(13,47)(14,37)(15,38)(16,45)(17,27)(21,32)(23,57)(25,52)(26,59)(28,51)(29,60)(30,56)(31,55)(33,36)(34,41)(40,58)(42,49)(44,64)(46,50)(48,63)(53,62)(54,61);(1,4)(2,10)(3,17)(5,27)(6,28)(7,25)(8,29)(9,34)(11,44)(12,46)(13,48)(14,23)(15,42)(16,49)(18,24)(19,53)(20,51)(21,52)(22,54)(26,32)(30,62)(31,61)(33,41)(35,58)(36,50)(37,63)(38,64)(39,47)(40,45)(43,57)(55,59)(56,60)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307205, #72307228, #72307258, … |
| maximal normal subgroup of | #72307205, #72307228, #72307258, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55148);