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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(7,11)(10,12)(13,14,18,24)(15,26,29,20)(16,30,33,22)(17,25,19,23)(21,35,40,32)(27,37,31,41)(28,38,34,42)(36,44,39,43);(2,3)(6,9)(7,11)(13,15,18,29)(14,20,24,26)(16,31,33,27)(17,34,19,28)(21,39,40,36)(22,41,30,37)(23,42,25,38)(32,43,35,44);(5,6)(7,10)(8,9)(11,12)(13,16,18,33)(14,21,24,40)(15,27,29,31)(17,32,19,35)(20,36,26,39)(22,23,30,25)(28,43,34,44)(37,42,41,38);(1,3,4,2)(6,9)(10,12)(13,16)(14,22)(15,27)(17,35)(18,33)(19,32)(20,37)(21,25)(23,40)(24,30)(26,41)(28,44)(29,31)(34,43)(36,38)(39,42);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,24,25)(15,28,29,34)(16,32,33,35)(20,38,26,42)(21,30,40,22)(27,43,31,44)(36,37,39,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7053145);