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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2001 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)(5,6)(7,9)(8,10)(11,12)(13,14,15,23,18,22,26,19)(16,27,28,38,30,36,24,20)(17,31,25,35,34,21,32,39)(29,44,42,37,33,43,41,40);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,23)(16,28,30,24)(17,32,34,25)(20,36,38,27)(21,39,31,35)(29,41,33,42)(37,43,40,44);(5,7)(6,9)(8,12)(10,11)(13,16,18,30)(14,20,22,38)(15,24,26,28)(17,33,34,29)(19,27,23,36)(21,40,31,37)(25,42,32,41)(35,44,39,43);(1,3,4,2)(7,12)(9,11)(13,17)(14,21)(15,25)(16,29)(18,34)(19,35)(20,37)(22,31)(23,39)(24,41)(26,32)(27,43)(28,42)(30,33)(36,44)(38,40);(6,10)(9,11)(14,22)(17,34)(19,23)(20,38)(25,32)(27,36)(29,33)(41,42)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6913745);