Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3516 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2940 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)(13,14,19,20,18,24,17,23)(15,25,31,21,28,38,27,35)(16,22,32,33,29,36,30,37)(26,39,42,34,40,44,41,43);(2,4)(5,6)(7,10)(8,9)(11,12)(14,20)(17,19)(21,25)(22,33)(23,24)(27,31)(30,32)(34,39)(35,38)(36,37)(41,42)(43,44);(6,9)(10,12)(13,15,18,28)(14,21,24,35)(16,26,29,40)(17,31,19,27)(20,25,23,38)(22,34,36,43)(30,42,32,41)(33,39,37,44);(6,9)(10,12)(13,16)(14,22)(15,26)(17,30)(18,29)(19,32)(20,33)(21,34)(23,37)(24,36)(25,39)(27,41)(28,40)(31,42)(35,43)(38,44);(5,7,8,11)(6,10,9,12)(16,29)(22,36)(26,40)(30,32)(33,37)(34,43)(39,44)(41,42)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6398714);