Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470927 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16717 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,28,7,13,6,12)(3,14,17,32,18,42,29,9)(4,10,20,34,21,35,26,38)(5,22,27,37,25,11,24,39)(15,43,50,30,46,40,45,58)(16,41,48,31,49,60,51,55)(19,52,23,56,53,33,54,36)(44,62,64,61,63,59,47,57);(1,3)(2,9,13,32)(4,15)(5,16)(6,17)(7,18)(8,29)(10,30,35,58)(11,31,22,55)(12,14,28,42)(19,44)(20,45)(21,46)(23,47)(24,48)(25,49)(26,50)(27,51)(33,57,52,61)(34,43,38,40)(36,59,56,62)(37,60,39,41)(53,63)(54,64);(1,4)(2,10)(3,15,18,46)(5,23)(6,26)(7,21)(8,20)(9,30,32,58)(11,36)(12,38)(13,35)(14,40,42,43)(16,47,49,64)(17,50,29,45)(19,24)(22,56)(25,54)(27,53)(28,34)(31,59,55,62)(33,37)(39,52)(41,61,60,57)(44,51,63,48);(1,5,7,25)(2,11,13,22)(3,16,18,49)(4,19,21,53)(6,27,8,24)(9,31,32,55)(10,33,35,52)(12,39,28,37)(14,41,42,60)(15,44,46,63)(17,51,29,48)(20,54,26,23)(30,57,58,61)(34,56,38,36)(40,59,43,62)(45,64,50,47)Polycyclic presentation:
2^9|a^2=efg,b^2=fh,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378124);