Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3149 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)(3,4)(7,11)(10,12)(13,14,19,23)(15,25,26,33)(16,22,30,38)(17,20,18,24)(21,28,36,29)(27,39,40,43)(31,34,32,37)(35,41,44,42);(2,4)(5,6)(7,10)(8,9)(11,12)(14,20)(15,26)(16,30)(17,18)(21,25)(22,37)(23,24)(33,36)(34,38)(35,43)(39,44)(41,42);(6,9)(10,12)(13,15,19,26)(14,21,23,36)(16,27,30,40)(17,29,18,28)(20,33,24,25)(22,35,38,44)(31,42,32,41)(34,43,37,39);(6,9)(10,12)(13,16)(14,22)(15,27)(17,31)(18,32)(19,30)(20,34)(21,35)(23,38)(24,37)(25,39)(26,40)(28,41)(29,42)(33,43)(36,44);(5,7,8,11)(6,10,9,12)(14,23)(16,30)(20,24)(21,36)(25,33)(27,40)(31,32)(41,42)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398331);