Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,23,19,17,18,24)(15,25,35,21,28,37,27,33)(16,22,30,40,32,34,31,39)(26,41,43,36,29,42,44,38);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,17,33)(16,29,32,26)(18,35,20,27)(22,38,34,36)(23,25,24,37)(30,43,31,44)(39,41,40,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,26)(17,34)(18,31)(19,32)(20,30)(21,36)(23,40)(24,39)(25,41)(27,44)(28,29)(33,38)(35,43)(37,42);(7,12)(9,11)(13,17,20,24,19,14,18,23)(15,25,35,21,28,37,27,33)(16,22,30,40,32,34,31,39)(26,42,43,38,29,41,44,36);(6,10)(9,11)(13,18,19,20)(14,23,17,24)(15,27,28,35)(16,30,32,31)(21,37,33,25)(22,39,34,40)(26,43,29,44)(36,41,38,42)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903862);