Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3486 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(14,19)(20,30)(21,31)(22,32)(33,41)(34,42)(35,36)(43,44);(6,9)(10,11)(14,19)(15,23)(21,31)(22,32)(24,37)(25,38)(35,36)(39,40);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,30)(24,33)(25,34)(26,35)(27,31)(28,36)(29,32)(37,41)(38,42)(39,43)(40,44);(5,7)(6,10)(8,12)(9,11)(13,15)(14,20)(16,24)(17,25)(18,23)(19,30)(21,33)(22,34)(26,39)(27,37)(28,40)(29,38)(31,41)(32,42)(35,43)(36,44);(13,16,18,27)(14,21,19,31)(15,24,23,37)(17,28,29,26)(20,33,30,41)(22,36,32,35)(25,40,38,39)(34,44,42,43);(13,17,18,29)(14,22,19,32)(15,25,23,38)(16,26,27,28)(20,34,30,42)(21,35,31,36)(24,39,37,40)(33,43,41,44)Polycyclic presentation:
2^9|a^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534623);