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 |
| prime radical | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,33,18,23,17,22)(15,24,26,20,28,35,27,36)(16,29,31,38,30,21,32,37)(25,40,43,44,41,39,42,34);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(16,30)(21,29)(25,41)(31,32)(34,44)(37,38)(39,40)(42,43);(5,7,8,12)(6,9,10,11)(13,15,18,28)(14,20,23,36)(16,25,30,41)(17,26,19,27)(21,34,29,44)(22,35,33,24)(31,42,32,43)(37,39,38,40);(2,4)(7,12)(9,11)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,34)(22,38)(23,29)(24,39)(26,43)(27,42)(28,41)(33,37)(35,40)(36,44);(6,10)(9,11)(13,17,18,19)(14,22,23,33)(15,26,28,27)(16,31,30,32)(20,35,36,24)(21,37,29,38)(25,42,41,43)(34,39,44,40)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 6969696);