Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3332 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,9,12)(7,11,8,10)(13,14)(15,19)(16,22)(17,20)(18,23)(21,25)(24,27)(26,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,27)(21,23)(22,28);(5,7,9,8)(6,10,12,11)(14,20)(16,17)(19,27)(21,28)(22,23)(24,26);(5,7,9,8)(6,11,12,10)(13,15)(14,21)(16,26)(17,24)(18,25)(19,23)(20,28)(22,27);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,22,23,20)(15,24,25,26)(19,27,21,28)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72283757 |
| maximal normal subgroup of | #72283757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307665);