Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,31)(24,29)(25,27)(26,39)(28,34)(30,38)(32,47)(33,46)(35,42)(36,50)(37,52)(40,41)(43,48)(44,51)(45,49);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,27)(24,35)(28,41)(30,39)(31,46)(32,36)(37,49)(48,51);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23,25,33)(22,28,27,41)(24,36,35,32)(26,40,38,34)(29,44,42,43)(30,46,39,31)(37,48,49,51)(45,47,52,50);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24)(22,29)(23,32)(25,35)(26,37)(27,42)(28,43)(30,45)(31,47)(33,36)(34,48)(38,49)(39,52)(40,51)(41,44)(46,50);(6,12)(7,14)(11,20)(13,15)(24,35)(29,42)(32,36)(37,49)(43,44)(45,52)(47,50)(48,51)Polycyclic presentation:
2^9|a^2=fhi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941957);