Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,26,30)(23,27,32,41)(24,35,36,29)(25,28,38,31)(33,47,45,40)(34,42,46,39)(37,44,49,43)(48,50,52,51);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,27)(24,33)(25,34)(26,32)(28,42)(29,40)(30,41)(31,39)(35,47)(36,45)(37,48)(38,46)(43,51)(44,50)(49,52);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(23,32)(25,38)(27,39)(29,43)(30,31)(33,45)(35,44)(37,49)(40,50)(41,42)(47,51);(7,15)(10,13)(14,20)(17,19)(21,24)(22,29)(23,33)(25,37)(26,36)(27,40)(28,43)(30,35)(31,44)(32,45)(34,48)(38,49)(39,50)(41,47)(42,51)(46,52);(1,3,4,2)(6,12)(10,13)(11,18)(17,19)(22,30)(23,32)(24,36)(28,31)(34,46)(37,49)(40,47)(50,51)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306259);