Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D3×#67 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C2×C32×Q8 | C22 |
| socle | C2×C32 | C22×D4 |
| hypercenter | C2×Q8 | D32 |
| hypocenter | C32 | #578 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×Q8 | |
526 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^2|b^2=f,c^2=f,d^2=f,e^2=f,c^a=cf,h^a=h^2,d^b=de,e^b=ef,g^b=g^2,d^c=df,e^d=ef60 conjugacy classes — view the full character table.
| maximal subgroup of | #11886684, #11886699, #11886703, … |
| maximal normal subgroup of | #11886684, #11886699, #11886703, … |
| maximal quotient of | #11858075, #11858078, #11858585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7210);