Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,17)(15,20,25,27)(16,21,26,18)(22,24,23,28);(2,3)(6,9)(10,12)(13,15)(14,20)(16,22)(17,27)(18,28)(19,25)(21,24)(23,26);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,22)(17,18)(19,26)(20,24)(23,25)(27,28);(1,3,4,2)(13,17,19,14)(15,23,25,22)(16,21,26,18)(20,28,27,24);(6,9)(10,12)(14,17)(16,26)(20,27)(22,23)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72256833, #72256861, #72256980, … |
| maximal normal subgroup of | #72256833, #72256861, #72256980, … |
| maximal quotient of | #72264684, #72264796 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467861);