Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
2180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,18,33)(16,26,29,34)(17,22,31,39)(21,27,37,24)(25,35,32,44)(28,42,30,43)(36,41,38,40);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,18)(14,20,23,33)(16,27,29,24)(17,25,31,32)(21,26,37,34)(22,35,39,44)(28,41,30,40)(36,42,38,43);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,24)(17,30)(18,27)(19,29)(20,34)(22,38)(23,37)(25,41)(26,33)(28,31)(32,40)(35,42)(36,39)(43,44);(2,4)(7,12)(9,11)(13,17)(14,22)(15,25)(16,28)(18,32)(19,31)(20,35)(21,36)(23,39)(24,40)(26,42)(27,41)(29,30)(33,44)(34,43)(37,38);(6,10)(9,11)(14,23)(16,29)(17,31)(20,33)(24,27)(25,32)(36,38)(42,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961331);