Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6)(7,19)(8,25)(9,31)(10,17)(11,16)(12,18)(13,22)(14,29)(15,34)(20,35)(21,49)(23,38)(24,37)(26,39)(27,55)(28,42)(30,41)(32,43)(33,44)(36,51)(40,56)(45,52)(46,65)(47,62)(48,63)(50,59)(53,58)(54,68)(57,60)(61,67)(64,66);(1,3,4,2)(5,7,10,22,12,24,11,23)(6,13,16,19,18,38,17,37)(8,26,28,20,30,47,29,48)(9,21,32,50,34,52,33,51)(14,39,41,63,42,62,25,35)(15,36,43,45,31,59,44,49)(27,54,56,46,58,66,57,67)(40,64,55,65,60,68,53,61);(2,3)(5,8,11,29,12,30,10,28)(6,14,17,25,18,42,16,41)(7,20,23,26,24,48,22,47)(9,27,33,57,34,58,32,56)(13,35,37,62,38,63,19,39)(15,40,44,53,31,60,43,55)(21,46,51,54,52,67,50,66)(36,61,49,68,59,65,45,64);(5,9,10,32,12,34,11,33)(6,15,16,43,18,31,17,44)(7,21,22,50,24,52,23,51)(8,27,28,56,30,58,29,57)(13,36,19,45,38,59,37,49)(14,40,41,55,42,60,25,53)(20,46,47,66,48,67,26,54)(35,61,39,64,63,65,62,68)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=ei,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304927);