Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394131 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #63090 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,33)(4,21,25,40)(5,26,29,38)(6,13,7,12)(9,19,34,18)(10,30,37,24)(11,27,39,28)(16,43,46,31)(17,47,50,32)(22,35,45,41)(23,36,55,60)(42,48,56,49)(44,59,64,61)(51,54,57,53)(52,58,63,62);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,33)(13,15)(14,34)(16,24)(21,31)(23,52)(25,45)(26,56)(27,49)(28,48)(29,50)(30,46)(36,59)(37,41)(38,42)(39,47)(40,43)(44,54)(51,58)(53,64)(55,63)(57,62)(60,61);(1,4)(2,10)(3,16)(5,23)(6,30)(7,24)(8,25)(9,31)(11,36)(12,40)(13,21)(14,37)(15,41)(17,44)(18,45)(19,22)(20,46)(26,57)(27,54)(28,53)(29,55)(32,58)(33,35)(34,43)(38,51)(39,60)(42,61)(47,62)(48,52)(49,63)(50,64)(56,59);(1,5)(2,11)(3,17)(4,23)(6,28)(7,27)(8,29)(9,32)(10,36)(12,26)(13,38)(14,39)(15,42)(16,44)(18,49)(19,48)(20,50)(21,51)(22,52)(24,54)(25,55)(30,53)(31,58)(33,56)(34,47)(35,59)(37,60)(40,57)(41,61)(43,62)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72264906, #72264949, #72265017, … |
| maximal normal subgroup of | #72264906, #72264949, #72265017, … |
| maximal quotient of | #72265220 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301328);